EP2425307A1 - Kinematic approximation algorithm having a ruled surface - Google Patents

Abstract

The present invention relates to a method for producing at least one surface on a workpiece by means of a material removal tool and a corresponding material removal device. The problem addressed by the present invention is to provide a method for producing any surface on a workpiece in such a way that the surface is produced quickly and at low cost. An error between any surface to be produced and a produced ruled surface should be small. The invention is characterized in that based on any surface to be produced, a movement path of the material removal tool is controlled to produce a ruled surface approximating to said surface to be produced, the movement path being provided in the form of a curve on a dual unit sphere, wherein a point on the curve corresponds to a location and an orientation of the material removal tool. The curve can be produced based on ruling lines, which are converted into points on the dual unit sphere by means of mathematical transformations. Said points are interpolated by means of a dual sphere spline interpolation algorithm to produce the curve. Said curve can then be transformed back into the ruled surface to be produced or can be used directly to follow the movement path of the material removal tool. Likewise, directrix curves of the ruled surface can be determined by means of the dual sphere spline interpolation algorithm. The method is suited in particular for producing the surfaces of components of turbomachinery, such as impellers. Any surfaces can be produced on any materials.

Description

description

Kinematic approximation algorithm with ruled surface

The present invention relates to a method for producing at least one surface on a workpiece by means of a material removal tool and a corresponding material removal device.

Workpieces may for example be components of industrial machines, in particular of turbomachines, such as propellers, impellers of centrifugal compressors, rotors of pumps, gas turbines or turbochargers. Workpieces can be general machined parts.

Conventionally, the geometry design part and the production part are separated. During the design phase, engineers construct a ruled surface and deliver the surface for fabrication. A ruled area may have approximated a free-form surface or been optimized according to design requirements. In the manufacturing phase certain processes for the production of the control surface are used. For example, five-axis flank milling involves the following steps: First, the router contact paths are generated from the input surface data. Then the

Receive cutter positioning data from the cutter contact data. Based on the cutter positioning data, motion sequences are planned for material removal tools. Finally, certain post-processing is used to obtain a numerical control code.

Unfortunately, the prior art has the following drawbacks: there is no global continuity guarantee, multiple optimization loops are required, loop computation is laborious, time consuming, tool positioning data can have local errors, and sufficient tool positioning data is required. It is an object of the present invention to provide a method for producing an arbitrary surface on a workpiece in such a way that the surface is produced quickly and inexpensively. An error between any surface to be generated and a generated control surface should be small.

The object is achieved by a method according to the main claim and a material removal device according to the independent claim.

A ruled surface is a surface that can be created by moving a straight line (straight line) in three-dimensional Euclidean space, so that a ruled surface can be created simply by removing material along a moving straight line Ruled surface can be called a ruling line, ruling, or ruling. The removal of material can be carried out, for example, by means of flank milling with a CNC (Computer Numerical Control) machine, electrical discharge wire cutting or laser cutting.

According to a first aspect of the present invention, a surface is formed on a workpiece by means of a material removal tool, controlling from a random surface to be generated a path of movement of the material removal tool to produce a control surface approximating the arbitrary surface, the path of movement being in the form of a curve on a dual unit ball, where a point on the curve corresponds to a location and orientation of the material removal tool. A smoothed one-parameter path representation with respect to a displacement of the tool can be provided. This is an accurate representation of the operation of a material

Removal device system. An analytical representation of the movement path of the material removal tool can be provided, so that a global error control for the Production is made possible. The theories of rule surfaces are combined with a screw theory and a binary number algebra. Using the algorithm, any given surface or discrete straight line sequence, so-called router positioning data, can be approximated by a rule surface. The arbitrary surface to be created may be provided as a free-form surface or as a discrete material removal tool positioning data. The arbitrary surface to be produced can be aerodynamically optimized, for example.

The advantages of a method according to the invention are a smooth single-parameter path representation through a curve on a dual unit sphere, in particular a dual-sphere splitting curve; a compact data structure for 5-axis milling in terms of position and orientation; Continuity and convexity; simple judgment as to whether the tool is in the working space or not; Real time; global error checking and small kinematic error; few cutting position data (CL-; cutter location data); suitable for various manufacturing processes.

According to a second aspect, a material removal device for carrying out a method according to the invention comprises a computing device, a control device and the material removal tool. The material removal tool is driven by the control means on the basis of the movement path calculated by the calculating means.

Further advantageous embodiments are claimed in conjunction with the subclaims.

According to an advantageous embodiment, the curve on the dual unit sphere can be defined as a continuous and smooth spin curve. The spline curve can be called a dual spherical spline curve. The curve on the dual unit sphere can be defined as a dual sphere spline. The continuity property of the spline avoids connection computation in the conventional motion path representation. The spline calculation algorithm is fast enough for real-time applications. A new type of spline is defined and called a "dual sphere spline". A ruled surface is represented as a dual ball spline on the dual unit sphere. This spline has advantageous properties in terms of continuity and convexity. A point on this spline corresponds to a position and orientation of a line in Euclidean space. The calculation of this spline is very fast, so a real-time requirement is met. This spline reduces the number of parameters to a third compared to conventional ruled surface parameterization techniques, such as the Tensor Product B-Spline surface.

According to a further advantageous embodiment, the following steps for providing the curve can take place:

Providing one of the arbitrary line of discrete line lines approximating the surface to be generated;

Transforming coordinates of a discrete linearization line in three-dimensional Euclidean space into coordinates of one discrete point on the dual unit sphere by means of a small-scale and then a study-mapping algorithm;

Interpolate the discrete points by generating the discrete points spline curve by applying a dual-sphere spline interpolation algorithm.

Based on the new type of splines, a series of algorithms for interpolating and calculating a dual sphere spline on the dual unit sphere are being developed. Consequently, a kinematic rule area approximation algorithm is developed.

According to a further advantageous embodiment, a Linierungsstraade the equation correspond; the dual-sphere spline interpolation algorithm has the following equations:

as an equation of the spline curve, where f _± basis functions and P ₁ control points can be on the dual unit sphere in ID- ^, with n (34) £ /, («) =!, /»> 0, Vi, where

weighted averages on the dual unit sphere may correspond to the following equation:

(17) q = Σ " ₌ o ^ω iP i ^{m with '} Σ G) ₁ = 1, 0) ₁ > 0, wherein to generate the spline curve, a minimization can be carried out according to the following formula:

According to a further advantageous embodiment, a calculation of the sequence of discrete line lines, which approximates the arbitrary surface, can be carried out by means of mathematical least-square minimization of distances to the arbitrary surface.

According to a further advantageous embodiment, the curve can be converted into the control surface in three-dimensional Euclidean space by means of an inverse study mapping algorithm and then an inverse small mapping algorithm. This conversion is not required if a material handling device can convert the data of the curve directly into a path of movement of the material removal tool. According to a further advantageous embodiment, the control points can be used as parameters for the approximation of the control surface to the arbitrary surface to be generated. A dual ball spline can be defined by a variety of control points. The spline can be determined by several control points. The control points can be used as parameters for optimization.

According to a further advantageous embodiment, the individual parameter u may be a feed rate or time with respect to a displacement of the material removal tool.

According to a further advantageous embodiment, the following steps can further be carried out to determine direct-error curves of the control surface determined on the basis of the linearization line:

starting from the arbitrary surface to be generated and the discrete line of linearization, in addition to each discrete line of linearization, a first and a second discrete reference line is determined, a first discrete reference line passing through an intersection of the discrete line of linearization with a first direct-line curve to be determined and a second reference line passing through an intersection of the straight line with a second direct-chemical curve to be determined, and the orientations of these reference lines correspond respectively to the surface normal of the surface to be generated at the intersections, where a distance of the two intersections of each discrete line of linearity corresponds to the length of the material removal tool;

- Transforming coordinates of a discrete reference straight line in three-dimensional Euclidean space in coordinates of one discrete point on the dual unit sphere by means of a small and then a study mapping algorithm, wherein the first reference straight sequence, a first discrete point sequence and the second reference straight sequence a second discrete dot sequence is generated; - interpolating the two discrete dot sequences by generating two further spline curves having the respective discrete dot sequences by applying a dual-ball spline interpolation algorithm; - Converting all three spline curves by means of an inverse study mapping algorithm and then an inverse Klein mapping algorithm in three rule surfaces in three-dimensional Euclidean space, wherein the two lines of intersection of the two rule surfaces of the first and second reference line respectively with the rule surface of the lining line the first and determine the second smooth and continuous direct-chemical curve of the rule surface of the linearization line, and the two direct-chemical curves through p (u) and q (u) of the equation

(2) x (u, υ) = (l-υ) p (u) + υq (u)

are described mathematically.

According to a further advantageous embodiment, a check can be made as to whether the movement path is within one

Working space of the material removal tool is, using the kinematic properties of a required movement and using a robotics analysis.

According to a further advantageous embodiment, the

Method for a shape design and a shape optimization can be used. Due to the reduction in the number of parameters compared to the prior art, the algorithm is preferably suitable for such use.

According to a further advantageous embodiment, the material removal tool can be part of a CNC (Computer Numerical Control) milling machine, an electrical discharge wire cutting machine or a laser cutting machine. According to a further advantageous embodiment, the workpiece may be part of a turbomachine, for example a propeller or a rotor.

The present invention will be described in more detail by means of exemplary embodiments in conjunction with the figures. Show it :

Figure 1 shows an embodiment of a control surface; Figure 2 shows an embodiment of a product with

Ruled surfaces; Figure 3 shows an embodiment of an inventive

process;

4a to 4d another embodiment of an inventive method.

FIG. 1 shows an exemplary embodiment of a control surface. A ruled surface is defined by the fact that the surface can be covered by moving a straight line (straight line) in the Euclidean space. Ruled surfaces are simple and inexpensive to manufacture. Ruled surfaces occur in many manufacturing processes.

A ruled surface is a preferred choice for manufacturing. A ruled surface is a special type of surface that can be created by moving a straight line in space. Ruled surfaces occur in various applications, such as electrical wire discharge machining (EDN) and laser cutting, which control the cutting tool as a moving straight line. In addition, it is known that a control surface can be effectively produced by using a flank milling method in CNC machining. To reduce manufacturing costs, it is a typical design strategy to approximate a free-form surface as a rule surface. Consequently, there is a need in the industry for an effective ruled surface approximation algorithm. A ruled surface is a simple object in geometric modeling. In the Euclidean space IR ^, a rule surface Φ has the following parametric representations:

x (u, υ) = a (u) + υr (u), «el, υe / i? (1)

Where a (u) is called the direct-chemical curve and r (u) is a generation vector. Alternatively, a control surface Φ can be parameterized by two direct-chemical curves p (u) and q (u):

The straight line designated by x (u _o , υ) = (l-υ) p (u _o ) + υq (u _o ) becomes

Called ruling straight line. A rule surface is a set of straight one-parameter lines.

Although ruled surfaces have been intensively studied in classical geometry, they are not fully utilized for geometric design and manufacturing applications. Concepts of Bezier curves and surface design were used to construct a ruled surface. The properties of ruled surfaces in the line geometry have been carefully studied. In straight line geometry, a rule surface is described as a curve in a quadrilateral in the P space. Based on these characteristics, according to the present invention, algorithms for interpolation and approximation of ruled surfaces have been developed.

Figure 2 shows an embodiment of a product with surfaces that can be approximated by ruled surfaces. Such surfaces can be, for example, surfaces of blades of a wheeled turbine. Other products may be, for example, propellers or turbochargers.

Fig. 3 shows an embodiment of a method according to the invention. There is provided a sequence of discrete line lines approximating the surface to be generated. This is followed by transforming coordinates each of a discrete line of linearity in the three dimensional Euclidean space in coordinates of one discrete point on the dual unit sphere using a small and then a study mapping algorithm. This is followed by interpolation of the discrete points by generating the dual-sphere spline curve having the discrete points by applying a dual-sphere spline interpolation algorithm:

Dual ball spline and its application in a rule surface approximation

Using dual numbers to represent straight lines (straight lines), a ruled surface is described as a curve on a dual unit sphere (DEK). A method of calculating weighted averages on the DEK based on minimized least squares minimization is shown. The presence, uniqueness, continuity and convexity of the weighted averages are discussed at the DEK. This leads to a novel definition of a dual ball spline on the DEK. A fast, iterative dual ball-spline interpolation algorithm is developed. Based on this algorithm, a kinematic ruled surface approximation algorithm is established which approximates a free-form surface with a ruled surface. This method can be used to design ruled surfaces and to approximate and schedule a motion path of a tool, such as a computer numerical control (CNC) machine.

Conventionally, the linear geometry in the kinematics together with the screw theory is used to describe geometric properties of the screw axis of a moving rigid body, which describes the manufacturing process for control surfaces. Using dual numbers, a ruled surface is described again as a curve on a dual unit sphere (DEK). The kinematically generated rule surface connects the path and the physical movement of the tool. An approximation algorithm based on this representation is not yet available. The key algorithm is based on the linear interpolation of a general dual quaternion. The goal is to approximate a given rule surface with a cylindrical tool motion curve.

In accordance with the present application, a new kinematic ruled surface approximation algorithm is introduced. This algorithm was developed based on the binary representation of a control surface. The problem of the approximation of the rule surface in Euclidean space is transformed into a curve approximation problem on the dual unit sphere. The difficulty of the curve approximation problem on the dual unit sphere is the nonlinearity of the space. Conventional linear interpolation methods are not applicable in the space of the dual unit sphere.

Starting from the definition of weighted mean in real sphere space, first a weighted average is defined on the dual unit sphere. The weighted average on the dual unit sphere is defined as a result of least squares minimization. This allows a novel method of defining Bezier and spline curves on the dual unit sphere. It could be proved that the problem of least squares minimization has a clear solution if the input points are on a dual hemisphere. The continuity and convexity properties of the dual sphere spline are also discussed. Based on these definitions, a kinematic ruled surface approximation algorithm is developed. The essence of this algorithm is a fast dual ball-spline interpolation algorithm on the dual unit sphere. This algorithm can be used to design surfaces in various fields, particularly for turbomachines such as propellers, centrifugal compressor impeller, gas turbine, and turbocharger. This algorithm can also be used to design the motion path and Planning the tool movement can be used for CNC machines.

In the following, the theoretical background of this approach is described. Furthermore, novel definitions of a weighted average on the dual unit sphere, a dual-sphere Bezier curve and a B-spline are proposed. In addition, an algorithm for calculating the weighted average on the dual unit sphere and a fast, iterative algorithm for the dual sphere spline

Interpolation provided. Furthermore, a kinematic ruled surface approximation algorithm for approximating a free-form surface with a ruled surface is proposed. Finally, a conclusion is made.

Theoretical background

Plücker coordinate of a straight line (straight line)

In the homogeneous Cartesian coordinate system, a straight

Line L are algebraically represented by two different points: X = (X ₀₅ X ₁₅ X ₂₅ X ₃ ) R = (X ₀₅ X) R and γ ^{on the} straight line:

i (t) = (l-t) x + tY (3)

Equivalently, a straight line L in the 3D projective space P can be represented by the outer product of two points X Λ Y, which is referred to as the homogeneous plücker vector coordinate LIR = (1, r) 1R:

(l, l °) = (x _o yy _o x, xxy) (4)

In Euclidean space, ie x ₀ = y ₀ = 1, the Plücker coordinates have a geometric interpolation, where l = r = y-x and l ° = xr = xy. These are the Plücker coordinates of an oriented straight line in E. Obviously, these co- Ordinatenelemente not independent. These coordinate elements fulfill the Plücker relation:

Ω _? (L) = I-Γ = O (5) The homogeneous Plücker coordinates 1,1 ° define a point in P. The length of the vector 1 is arbitrary and can be standardized:

1-1 = 1 (6) Not every point in P is a Plücker coordinate. Only the points that satisfy the Plücker equation Equation 5 are Plücker coordinates. Equation 5 defines a quadratic variety in P, which is referred to as small quadrant M ₂ . In this way, the bijection map γ: L → M _{2 can be established} between straight lines L e P 3 and points LIR e M ₂ 4. This picture is called "Map" or "Picture".

Dual number representation of a straight line (straight line)

A straight line can also be represented in a more compact manner using binary numbers. A binary number can be written in the form a = a + εa °, where a, a ° e is IR and ε is the dual element with ε = 0. Dual numbers can be extended into the vector space, the space ID is defined as a set of all pairs of vectors:

a = a + <sa ° if a, a ° e IR ³ (7)

If two dual vectors, x = x + εx ^° and y = y + <sy ^° , then the inner product in ID is defined as follows:

i-y = x-y + ε (x ° -y + x-y °) (8)

Thus, the length of a dual vector is defined as

A dual vector of length 1 is called a dual unit vector. Obviously, a dual unit vector satisfies the following equations:

With reference to Equations 5 and 6, it is possible to obtain a more compact representation of a straight line: The binary number representation of a straight line is simply the writing of the student coordinates as a dual unit vector. The computation problem of computing points on a quad in P is reduced to a task in a dual form of spherical geometry. This illustration or map is referred to as a study map or study map.

Dual number representation of a rule surface

Dual unit vectors define points on a sphere in ID. This ball is called a dual unit ball (DEK). In this form, a rule surface defined by Equation 1 is written as a curve on the dual unit sphere:

A binary number representation of a rule surface can be converted into an algebraic form:

x (u, v) = l (u) ^° + v \ (u) (12) Now, a transformation mapping between a rule plane representation in Euclidean space and a curve representation on the dual unit sphere is established. Instead of solving a surface approximation problem in Euclidean space, a curve approach problem is solved on the dual unit s sphere.

Dual ball splice

Weighted mean and spline on a real ball.

A weighted average on a real sphere is beru ^¬ starting defined on minimizing, according to the method of least squares. Let P ₁ , ..., p _{n be} points on a d-dimensional emusivity sphere 5 in IR, a weighted average of these n-points use weighting values COi, ..., ^ω n such that each CO ₁ > 0 and ^ &> _] _ = 1, the weighting

1 mean is referred to as:

C = Σ _{1 = 0} CO ₁ P ₁ (13)

The weighted average in Equation 13 is not simply a linear combination of the points p _lr ..., p _n , but a result of least-squares minimization, which is less than point C to 5, which minimizes the following value :

^f ( ^c ) = ^■ dist _s {c, P ₁ ) ^{2 (} 14 ⁾

where dist _s {c, P ₁ ) is the sphere distance between C and P ₁ .

The function f reaches a unique minimum if the following condition is met:

Theorem 1. Suppose the points P ₁ , ..., p _n are all in a hemisphere H of 5, with at least one point P ₁ in the interior of H with CO ₁ ≠ 0. Then the function f A single critical point C in H, where this point C is the global minimum of f.

It can be shown that the newly defined weighted average has advantageous properties. Based on the definition of a weighted average on a real sphere, the spline functions taking values on the unit d-sphere 5 can be defined analogously. Now let p _lr ..., p _{n be} the points on 5 and let Jf ₁ (u), ..., f _n (u) be basis functions that satisfy the following property: i = l

for u in the interval [a, b]. The spline curve s (u), which takes values on the unit sphere, is defined as:

s (u) = Σ? _{= 1} f ₂ (ü) p ₂ (16)

The most common applications of splines verwen- the B-splines with the basic functions Jf ₁ (u), the portion ^¬ as cubic curves with continuous second derivatives are. From a continuity theorem it is known that if the basis functions f _{Σ have} continuous kt derivatives, the spline curve also has kt derivatives. In this case, the Kugelsplinepunkte s (t) are sufficiently defined, provided that each of four aufeinanderfol ^¬ constricting control points (control points) is located in a hemisphere.

Weighted average (weighted average) on the dual A ^¬ ness ball

The principle of transfer of dual unit vectors simply means that for every operation that is defined for a real vector space, there is a dual version with the same interpretation. Based on this transference In the case of dual unit vectors, a similar definition of a weighted average on the dual unit sphere can be derived. Since only the case for the dual unit sphere in ID is of interest, the definition can be narrowed down as follows:

Definition 1. Let pj, ..., p _n on the dual unit sphere

5 in ID. A weighted average of these n points using real weighting values ωγ, ..., ω _n such that each CO ₁ > 0 and Σ _ι ω _ι = \, the weighted average of these n points is given as:

It is defined as a result of least squares minimization, namely as the point q on

5, which minimizes the following value:

in which is.

The distance between two points on the dual unit sphere is defined by a dual angle between two straight lines. It has the form θ = θ + ε -d, where θ is the angle between the straight lines and d is the minimum distance along the common vertical. For two points x and y on the dual unit sphere, the following equation results:

x -y = cos / 9 (19) The dual arc cosine function is defined as:

θ = COs ^{- 1} Jx + £ x °) = (20)

Similarly, the theorem is for the existence and uniqueness of the definition. Theorem 2: Assume that the points pj, ...., p _{n are} all on a dual hemisphere H of 5, with at least one point p {inside H with CO ₁ ≠ 0. Then the function f a single critical point q in H, this point q being the global minimum of f.

Existence and uniqueness

Theorem 2 is proved. Before the proof, the exponential and logarithmic functions for the dual vectors are defined. These functions are useful for the proof and for the development of the algorithm.

Exponential and logarithmic functions

First a subspace of ID is defined, which is represented as follows:

r ^ {i | i = (jc ₁ , Jc ₂ , θ) tJc ₁ , Jc ₂ e / D} (21)

Obviously, the subspace T is a linear space. The norm defined in equation 9 is still valid for calculating the distance between two points in the subspace. This subspace can be assumed to be a tangential hyperplane with respect to a point q on the dual unit sphere. Without limiting the general validity, a point q: = (0,0,1) is selected, whereby the points on the tangential plane of point q can be written as Xi = (X ₁₅ X ₂₅ I). Assuming that the point q is the origin of T _q , one obtains exactly the subspace T. Then the distance between q and p \ on the hyperplane can be calculated as follows:

r = x- M = U ^{, χ} ₂ M (22) The exponential imaging at q is defined by the Ab ^¬ form of points _q of the tangential hyperplane T to the dual unit ball, which holds beibe ^¬ angles and distances of q. The exponential mapping is called expq (.). In this case, a function is given which places a point p with the coordinates (X ₁ , £ 2,1) at a point exp, _| (p) = (x [, x ₂ ', x ₃ ).

The following condition should be true to maintain the distance:

x ₃ = cos (f) (23)

where f is defined by Equation 22. Since exp ^ (p) is located on the dual unit sphere, assuming the property sin (r) + cos (r) = 1, the following is defined:

~, ~ sin (f) _Λ , _Λ sin (f) x = X ₁ ^■ ^ andx ₂ = x ₂ ^• r ^ (24) rr

In case r = 0, the dual divisor is not defined, so that X ₁ = X ₁ and x ^" 2 = X2 are assigned.

The logarithmic function is the inverse function of the exponential mapping, which points to a point P '= (xj, X2, X3) on the du- ring unitary sphere maps to the tangen ^¬ tialen hyperplane T _Q, provided that p 'is not antipodal to q. We denote the logarithmic function as lq (.) And expq (lq (p ')) = p'. Consequently, the reversal ^¬ figure is defined as follows:

where θ = cos (St ₃ ) is the dual angle between p 'and q. It is assumed that the main part of θ is the following equation satisfied: 0 <θ <π. In the case of θ = 0, St ₁ = St ₁ for i = 1, 2.

Proof of presence

Since f is a continuous function on the compact space 5, f attains its minimum value at least at the point q. It can be proved that q lies inside the hemisphere H.

Assuming that q is the minimum of Eq. 18 and completely outside H, a point q 'in the

Inside of H, by reflecting q through the edge of H. Obviously, for the point Pj within the hemisphere, the distance between Pj and q 'is smaller than the distance between Pj and q. For points Pj on the edge of H, the distances are the same. The value of / Xq ') </ (q) contradicts the assumption. Therefore, the minimum q can not be outside H.

Next, it is proved that the minimum q also can not be on the edge of H. It is equivalent to prove that the gradient of f at the edge is always nonzero and points outside H. Using the maps used above, F (s) = / (exp, _j (s)) for the points S on the tangential hyperplane T ^. The axes X ₁ , x _{2 are selected} for T- and then the first derivatives

from f to q _~ (ldτλ is defined. The best description of the ab- line of f is gradient vector Vf, this is a tangent vector to the dual sphere at q:

(26) where O ₁ and U _{2 are} the unit vectors aligned in the direction of the axes St ₁ and St ₂ . For the uniqueness proof, it is necessary to verify that the second derivative of f at point q is positive. Its second derivatives of q are equal to (S ² F). For the remainder of the proof

OX ₁ OX _j is used for bolt calculations.

screw theory

In terms of stiff motion, a screw is a way to describe a displacement. The displacement may be thought of as rotation about an axis and translation along the same axis. A general screw S consists of two parts, a true 3-vector S indicating the direction of the screw, and a true 3-vector

P, which, by recording the moment of the screw around the

Origin, S isolated. In this regard, a screw is represented as a dual vector:

S = S + ε S ^ = S + ε (pS + S ₀ ) (27)

where p is the thread pitch "pitch" of the screw and SQ is the moment of the line of the screw around the origin. SQ derived from the origin of the radius vector R, or more generally from any point V of the ball with SQ = RxS = Vxs. S ₀ is at right angles to S (SS ₀ = Oj.

Obviously, a straight line is a screw where the thread is 0, ie p = 0. Therefore, one can use screw calculations to analyze the partial derivatives of the function f at the point q on the dual unit sphere, f being a dual scalar function of the screw S, which has the following form: (28) The dual vectors argument is expressed in a right angle coordinate system with origin 0, and then the formulas are applied to a dual number argument. The dual coordinates of the screw are as follows:

where S _x , S _y , s _z , S _x , s _y °, s _z ° are six real elements of a Plücker coordinate. Using the differential rules for binary functions, we get:

The function f becomes real if all variables are real, so f (s _x , s _y , s _z ) = f (s _x , s _y , s _z ). After conversion to vector notation, the following are obtained:

f (s) = f (a) + εs ° ^■ Wf (s) = f (a) + ε (s ° ^■ v) f (s) (31)

When the above equation is analyzed, it can be seen that the screw function f (S) is completely defined by a function of its main part f {s). Consequently, the following property results: It is known that two dual vector functions F (x) and Φ (x) satisfy the following equation:

VF (x) = φ (x) (32)

The following identity can be inferred:

VF (x) = Φ (x) (33)

To prove that the gradient of f at the edge is always nonzero and directed to the outside of H, it is equivalent to prove that the real vector function f (x) at Edge is always non-zero and is directed to the outside of the real hemisphere H. This has already been done in the literature. More generally, the following theorem is derived:

Theorem 3. All formulas and all theorems of vector analysis remain in force in the field of screws.

It follows from the above that a screw analysis can be formed by replacing screws in vectors. Obviously, the relationship between geometrical objects explained above is retained: the dual modulus of the screw corresponds to the magnitude of the vector, and the dual angle between the axes of the screws corresponds to the angle between vectors.

Instead of proving that the dual vector function f (x) has a unique minimum, it proves that the principal part of f (x) has a unique minimum, followed by exactly the same procedure as the proof for the uniqueness of a weighted average on the real sphere.The evidence is not repeated, the details can be found in the relevant literature.

Continuity and convexity properties

It has been proved that the derivative property of a screw function is completely determined by its main part. Consequently, the same continuity theorem as in the case of the real sphere results:

Theorem 4. Let the values for P ₁ , ..., p _n and ω ^, ..., ω _n and q be chosen such that they satisfy the hypotheses of Theorem 2. The result is a neighborhood of P ₁ , ..., p _n , a> i, ..., a> _n , where the weighted mean q is a C ^∞ function of P ₁ , ..., p _n , ω ^, ..., ω _n , is. It can also be shown that the points q, which can be written as a weighted average of P ₁ ..., p _k , produce a convex set. They produce exactly the convex surface of the points P ₁ , ..., p _k .

Definition of dual sphere splmes

Based on the definition of a weighted average on the dual unit spheres in ID ^, the spline functions assuming values on the dual emusivity sphere can be defined analogously. Just as with the definition of splices on a real sphere, the base functions must always satisfy the following property:

Σf _ι (u) = \, f _ι (u) ≥0, Vi (34) i 1

For u in the interval [a, b].

Since Bernstein polynomials and B-splice basis functions both satisfy this requirement, the dual sphere Beziers

Curve or B-spline curve s (t) defined as follows, the

Accept values on the dual unit sphere:

a (u) = Σ ⁿ _{1 = 1 2} f (u) ρ ₂ (35) P _1, ..., p _n are the points on the unit sphere in the dual ID ^ -

To satisfy the uniqueness requirement, for each value of the parameter u, the set of control points p ₂ for which f _j _ {u) ≠ 0 is contained within a dual hemisphere. At a minimum, each value is usually contained within a hemisphere to satisfy the uniqueness conditions.

Interpolation of dual sphere splmes Algorithm for calculating weighted averages on the dual unit sphere.

A new algorithm for calculating the weighted average on the dual emusivity sphere is proposed below. The basic idea of this algorithm is to use the logarithmic mapping that maps all the points P ₂ on the dual unit sphere onto the tangential hyperplane to q, then calculate their weighted mean in the hyperplane and return that result to the dual unit sphere through the exponential mapping maps. The exponential mapping is defined according to equations (23) and (24). The logarithmic mapping is defined according to equation (25). All calculation rules are based on the calculation rules defined in the dual vector space ID ^.

Since the exponential mapping and the logarithmic mapping are defined only at q = (0,0,1), for a general point q on the dual unit sphere, the coordinate frame must be moved so as to match the mapping. The matrix for moving a point X ₁ to a point X ₂ on the dual unit sphere is given by the formula:

X ₂ = [£] * l (36) where [R] = exp (ά [adg]) (37)

= [l] + sm (ωfadg] + (cos (ω) - l) ([adg]) ² ώ corresponds to the dual angle between the points X ₁ and X ₂ :

X ₁ • x ₂ = cos ώ = x + εx ° (38) and the screw axis g is chosen so that it is perpendicular to both points X ₁ and x ₂ : g = ^^ f (39)

The result is the following algorithm:

Algorithm for calculating weighted averages on the dual unit sphere.

• Input: P ₁ , ..., p _n on the dual emusivity sphere and non-negative direction factors ω ^, ... ω _n with the sum

1; • Output: the weighted average of the input values;

• Initialization: Set q: = ^ ω ₂ p ₂ / Σ _-, ®iP _j

• main loop: ^for i = 1; ...; n 'set pl _{: = l} ~ (pj, set u: = Σ "^ (p) -q)' set q _{: =} exp ^ (q + u) 'if the principal value of | k _j || is sufficiently small, output q and stop, otherwise continue looping.

Here, i ~ (p) the mapping that maps points £> to the tangential hyperplane at q and χp _e ~ (q + u) is the result ^¬ He returned to the dual unit sphere.

Algorithm for splme interpolation on the dual unit sphere

Using the dual sphere splme defined in equation (35), the dual sphere-splme interpolation problem can be solved. On the basis of points given _C1, ..., c ^{au he} fd dual unit sphere and starting from Pa ^¬ rametern _u <u ₉ <<u want 'a Glattungskurve on the dual unit sphere are found, which is by _u parametπ- Siert in such that s (u) = c ^ ^{ur a} H ^e ± • The fundamental problem is the choice of additional node positions. and control points p, which define a ball-spline curve according to equation (35) and satisfy these conditions. Here f ( _u \ is chosen as cubic B-spline basis functions and an iterative method is used to solve for the control points p, which can easily be extended to B-splines of higher order.

By definition, there are _{n +} 2 control points for _n input points. Let P _{1 =} p ₂ and ^ _{n + 1} = ^ _{+ 2} , where a _lf ß _{lf rΣ denote} the elements in a base matrix that are not 0.

Bas i smatrix •

(40)

The dual ball-cubic B-Spine interpolation algorithm can be described as follows:

Algorithm for the interpolation of dual ball-cubic B-

Splines:

• input: points £ • • £ and real coefficients Ci ₁ , β _lf Y ₁ , (o ≤ i ≤ n);

• control points fj;

• initialization: set fj - ₌ g for j_ ₌ ] _ _M ;

• main loop: f ^or i = 1; ...; n, set ^; = _ («. /.(p) ₊ ^ ./-(p _{! +2} )> ß _t . set S ₁ = Ip ₁ -I ^ l _' put set P ₁ = P ₂ and P _{n + 2} = P _{n +} I '

if the sum of the main values § ■ of § ■ is sufficiently small

5 is stopped; otherwise the loop pass is continued.

When the control points are derived, the dual ball spline is the weighted average of the control points

] _0 calculate. The running time of the weighted average algorithm is an order of magnitude smaller than the running time of the interpolation algorithm, so that the time required to calculate a large number of points along the curve is the time required to calculate the control points.

] _5.

simulation results

The algorithm is tested with different input values. The input Especially sequence is given in the form of dual vectors \ - = \ - + gf ι wherein j_ ₌ i _r ... _r n • ^E i corresponding to ⁿ point on the dual unit sphere of an infinite straight line in Euclidean space.. The dual vector representation of straight lines is transformed into the algebraic representation of straight lines in order to display the input sequence of frames:

I ₁ (V) = I ₁ Xl? + V l ₁ , for i = 1,. , , , n (4 1)

v can be an element of the range [0,1]. To apply the _3Q interpolation algorithm for dual ball-cubic B-splines, the parameter and node sequence must be determined. The chord length was chosen to define the parameters: Let J be the chord length between two given points

J ₌ J .J z = l ... n ', ^so the total chord length is calculated by 35 ^ _ y ^ ⁿ rf. Since ^ is a dual number, the Body of ^ ^a l ^s d is used and the parameters are calculated as follows:

U ₀ = 0 (42)

U ₁ = u _l _ _l + -, ι = l, ..., nl

U _n = 1

This dual ball-spline allows the use of beliebi ^¬ gen account items. For simplicity, the node sequence is selected according to the parameters.

The algorithm converges quickly and the interpolation ^¬ error is low. The end result is given as a cubic B-spline Dualku ^¬ gel:

s (u) = Σ _i ⁿ _{= 1} f _i (u) p _i which satisfies the condition s («) = ϊ. It may be the dual-specific kubi ^¬ B-Spline be represented as a ruled surface, given by equation 12, where _L> e [θ, ll is. It can be the input odds sequence and the points on the interpolated

Spline, given by _$ ( _u \) The algorithm could be verified.

Kinematic Ruled Surface Approach and its application

The Dualkugelsplineinterpolationsalgorithmus can be used to Annähe ^¬ tion of a given free-form surface with a ruled surface. For the ruled surface approximation algorithm, the first step is to find a discrete system of rulings near the given surface. ^Subsequently , this linear line of linear gradation derived in the first step is written in the form of dual vectors corresponding to points on the dual unit sphere. Then the dual sphere-spline interpolation algorithm can be used to derive a cubic B-spline curve be applied to the dual unit sphere, which corresponds to a rule surface in the Euclidean space. This curve can be mapped back to a rule surface in Euclidean space based on Equation 12. Two Directrix curves 5 on a ruled surface can be written as follows:

10 q (M) = 1 (M) x1 (M) ⁰ HU ₂ I (M) (43b)

This representation contains two additional parameters _υ and _υ , so that additional information is needed to determine the edges of the control surface. For _different] _5 applications, a variety of methods can be applied. Here, a kinematic ruled surface approximation algorithm suitable for, for example, designing and manufacturing centrifugal compressor blades is proposed.

20

In straight line geometry, a point can be interpreted as an interface of two straight lines. A point on the edge of a ruled surface is defined by cutting a ruling line with a reference line. More precisely, the reference

25 just defined by passing through a point on a directrix curve and aligning the reference line with the surface normal at that point. This definition of the reference line is inspired by the manufacturing process, with the line of line

3 _Q (ruling), the normal of the surface and a unit vector perpendicular to the line of ruling and the normal form a local coordinate system for the moving milling tool. Obviously, the movement of the reference straight also produces a rule surface. Therefore, the Du-

35 al-ball spline interpolation algorithm using the reference line as input. A directrix curve is created by cutting these two ruled surfaces. passes. Likewise, the other directrix curve can be derived by repeating the above procedure.

The following steps can be carried out to determine direct-edged curves of the rule surface determined by the ruling line:

a first and a second discrete reference straight line are determined, starting from the arbitrary surface to be generated and the discrete linearization straight line, a first discrete reference straight line passing through an intersection of the discrete linearizing line with a first directivity curve to be determined and a first discrete reference straight line the second reference straight line passes through an intersection of the linear line with a second direct chemical curve to be determined, and the orientations of said reference line correspond to the surface normal of the surface to be generated at the intersections, wherein a distance of the two intersections of each discrete line straight corresponds to the length of the material removal tool; - Transforming coordinates of a discrete reference straight line in three-dimensional Euclidean space in coordinates of one discrete point on the dual unit sphere by means of a small and then a study mapping algorithm, wherein the first reference straight sequence, a first discrete point sequence and the second

Reference straight sequence a second discrete dot sequence is generated;

- interpolating the two discrete point sequences by generating two further spline curves having the respective discrete point sequences by applying a dual-sphere spline interpolation algorithm;

- Converting all three spline curves by means of an inverse study mapping algorithm and then an inverse Klein mapping algorithm in three rule surfaces in three-dimensional Euclidean space, wherein the two lines of intersection of the two rule surfaces of the first and second reference line respectively with the rule surface of the lining line the first and second smooth and continuous direct curve of the control surface of the Determine line of linearity and where the two direct-chemical curves are given by p (u) and q (u) of the equation

x (u, υ) = (l-υ) p (u) + υq (u) (2)

5 are described mathematically.

In other words, the two direct radicals of the control surface associated with the line of lines can be determined as follows, for example. A framework for the kinematic ruled surface approximation algorithm is obtained:

Step Sl. Extracting the line of linearity from the given area and determining the reference line corresponding to _] _5 of two directrix curves;

Step S2. Transforming the coordinates of the three straight line sequences into the coordinates of the points on the dual unit sphere ^; Transforming Coordinates of a Discrete 20 Straight Line in Three-Dimensional Euclidean Space into Coordinates of a Discrete Point on the Dual Unit Sphere Using a Small and Then a Study Mapping Algorithm;

Step S3. Applying the Dual-Ball B-Spline Algorithm;

Step S4. Calculating the dual-sphere B-spline using the dual-sphere weighted average algorithm;

_3Q step S5. Transforming the dual-number representation of the rule surface back into the Euclidean space; Curves on the dual unit sphere can be transformed into a rule surface in the three-dimensional Euclidean scheme using an inverse study mapping algorithm and then an inverse small-mapping algorithm.

35 see space to be transformed.

Step S6. Determine the two Directrix curves by cutting ruled surfaces. FIG. 3 shows the sequence of steps for determining a control surface which has been approximated to any desired surface to be produced.

5

For example, this algorithm was used to design

Used blade surfaces. For verification of the algorithm, a centrifugal compressor blade was selected which is approximately designed to a control surface than the input for

] _0 the algorithm. A simulation result for the kinematic spherical surface approximation algorithm was obtained. It can be represented the original shape of the given blade. It is possible to extract three sequences of straight lines which form a group of, approximate, the given blade.

15, straight lines and two groups of normals. These three groups of straight line sequences can be represented. Using the dual ball spline interpolation algorithm and the straight line intersection algorithm, an approximate control surface is derived. It can use the approximate ruled surface

20 are compared to the originally given blade area, showing that the approach error is very small. This ruled surface is represented by a straight line path that creates the surface so that a close connection to the manufacturing process is given.

25

conclusion

It has been described above how a ruled surface approximation problem in Euclidean space is transformed into a curve

3 _Q interpolation problem was transformed on the dual unit sphere by applying a Klein Map Algorithm and a Study Map Algorithm. A weighted average has been defined on the dual unit sphere that leads to the definition of a dual sphere spline on the dual unit sphere.

35 ball leads. Based on these definitions, fast, iterative algorithms are proposed for the calculation of weighted averages and interpolation of dual sphere splits on the dual unit sphere. These algorithms are defined by various inputs and are extended to a kinematic ruled surface approximation algorithm. This novel ruled surface approximation algorithm can be used to approximate a freeform surface with a rule surface. This can be used to design surfaces and to plan toolpaths, for example on CNC machines. Therefore, the kinematic ripple approximation algorithm has a high value for industrial production and has many application possibilities in various fields. Any surface can be created on any material.

A method according to the main claim is sufficient for a workpiece machining, since the tool only a

15 has a certain length and in this way a control surface is generated. In addition, according to the invention, the direct-chemical curves can be determined. Furthermore, a material processing device can directly use the data of the dual-sphere spline curve to generate a control surface. A mate

20 rial processing after conversion into a rule surface in the Euclidean space is also possible. The arbitrary surface to be generated can be aerodynamically optimized, determined by structural data, determined by an experiment or determined by other criteria. It can become one

25 curved surface arise.

Figures 4a to 4d show a further embodiment of a method according to the invention. 4a to d show the control of a flank milling device by means of computer numerical

3 _Q-driven control (CNC, computer numerical control). FIG. 4a shows, in a first step, a lower surface to be generated and an offset surface. 4b shows in a second step the discrete positions of the material removal tool. 4c shows in a third step the movement

35 material removal tools and the surface produced. Fig. 4d shows a comparison between the manufactured surface and a given blade to be produced as the surface to be produced.

Claims

claims A method of producing an area on a workpiece by means of a material removal tool, wherein a path of movement of the material removal tool to produce a control surface approximating any surface is controlled from any surface to be generated, the movement path being in the form of a curve on a dual unit sphere with a point on the curve corresponding to a location and orientation of the material removal tool. 2. The method according to claim 1, characterized in that the curve is defined on the dual unit sphere as a continuous and smooth Dualkugelsplinekurve. 3. The method according to claim 2, characterized by Providing one of the arbitrary line of discrete line lines approximating the surface to be generated; - transforming coordinates of a discrete linearization line in three-dimensional Euclidean space into coordinates of one discrete point on the dual unit sphere by means of a small-scale and then a study-mapping algorithm; Interpolate the discrete points by generating the dual spherical-spline curve having the discrete points by applying a dual-sphere spline interpolation algorithm. 4. The method according to claim 3, characterized in that a Linierungsstraade the equation corresponds; the dual-sphere spline interpolation algorithm has the following equations: S (Ii) = Z ₁ ¹ UZ (Ii) P ₁ , (35) as the equation of the dual-sphere spline curve, where f _x basis functions and p _; Control points on the dual unit sphere in ID 3 are, with where (34) weighted averages on the dual unit sphere correspond to the following equation: q = Σ " _{= o} ω _ι p _ι with Σω _t = l, ω _t > 0, wherein (17) to generate the dual-sphere spline curve, a minimization is carried out according to the following formula: / (q) = ^ Σ ^ - ^ (q, p _! ) ^{2 (} i8 ⁾ . Method according to claim 3 or 4, characterized in that a calculation of the sequence of discrete line lines approximating the arbitrary surface is carried out by means of mathematical least-square minimization of distances to the arbitrary surface. 6. The method according to any one of the preceding claims, characterized in that the curve is converted by means of an inverse study mapping algorithm and then an inverse small-mapping algorithm in the rule surface in three-dimensional Euclidean space. 7. The method according to claim 4, characterized in that the control points are used as parameters for the approximation of the control surface to the arbitrary surface to be generated. 8. The method according to claim 4, characterized in that the individual parameter is u feed rate or time with respect to a displacement of the material removal tool. 9. The method according to any one of the preceding claims 3 to 8, characterized in that a first and a second discrete reference straight line are determined, starting from the arbitrary surface to be generated and the discrete linearization straight line, a first discrete reference straight line passing through an intersection of the discrete linearizing line with a first directivity curve to be determined and a first discrete reference straight line the second reference straight line passes through an intersection of the linear line with a second direct chemical curve to be determined, and the orientations of said reference line correspond to the surface normal of the surface to be generated at the intersections, wherein a distance of the two intersections of each discrete line straight corresponds to the length of the material removal tool; - Transforming coordinates of a discrete reference straight line in three-dimensional Euclidean space in coordinates of one discrete point on the dual unit sphere by means of a small and then a study mapping algorithm, wherein the first reference straight sequence, a first discrete point sequence and the second Reference straight sequence a second discrete dot sequence is generated; - interpolating the two discrete point sequences by generating two further dual spherical spline curves having the respective discrete point sequences by applying a dual-sphere spline interpolation algorithm; Convert all three dual-sphere spline curves using an inverse study mapping algorithm and then an inverse one Small mapping algorithm in three ruled surfaces in the three-dimensional Euclidean space, wherein the two lines of intersection of the two ruled surfaces of the first and second reference lines respectively with the ruled surface of the Linierungsgeraden determine the first and second smooth and continuous direct-electron curve of the rule surface of the Linierungsgeraden, and wherein the two direct-eddy curves by p (u) and q (u) of the equation x (u, υ) = (l-υ) p (u) + υq (u) (2) are described mathematically. A method according to any one of the preceding claims, characterized by checking whether the path of movement is within a working space of the material removal tool, using the kinematic properties of a required movement and applying a robotics analysis. 11. The method according to any one of the preceding claims, characterized in that the method for a shape design and / or a shape optimization of the workpiece is used. 12. The method according to any one of the preceding claims, characterized in that Material removal tool is part of a CNC (Computer Numerical Control) milling machine, an electric discharge wire cutting machine or a laser cutting machine. 13. The method according to any one of the preceding claims, characterized in that the workpiece is a component of a turbomachine, for example, a propeller or a rotor. 14. A device for carrying out a method according to one of the preceding claims, characterized in that a control means drives a material removal tool according to a method of any one of the preceding claims, wherein a calculation means calculates a movement path of the material removal tool.